User:M.R.Forrester/Cubic functions

Copied from WP:REFDESK/MATHS, 11 January 2009

Cubic functions

I'm trying to use cubic (and similar) functions for a little model, but I really don't have much maths. Could a mathematician please explain whether one changes a, b or c to make the curves flatter/steeper, change the y intercept, etc.? The article on quadratic equations does this graphically through a clever image, but a few lines of text would be great. I'd like to add this info to the relevant article as well. --Matt's talk 14:09, 11 January 2009 (UTC) Edited to clarify which article is relevant --Matt's talk 14:17, 11 January 2009 (UTC)

I presume you mean that the cubic is of the form:

$f(x)=ax^{3}+bx^{2}+cx+d\,$

In that case, "flatness" of the curve is measured by its derivative which is (at x):

$f'(x)=3ax^{2}+2bx+c\,$

So only the coefficients a, b and c have an impact on the steepness of the curve (the greater these values are, the greater the steepness; the smaller these values are, the greater the flatness). The y-intercept is given by the image of 0 under f so the value of d equals the y-intercept. If d is 0, the curve passes through the origin. PST

The article elliptic curve might also be of interest to you. PST

And by the way, mathematicians usually use one branch of mathematics in another branch of mathematics. There are numerous examples of this (I might as well let someone else list these examples; there are so many that I can't be bothered!). One interesting example is applying graph theory and the theory of covering maps to prove the well known Nielson-Schreier theorem; i.e every subgroup of a free group is free.

On the same note, there are mathematicians who would prefer not applying mathematics to another field (theoretical mathematicians) and those who would prefer applying mathematics to another field (applied mathematician). From experience, applied mathematicians are generally not so interested in the theoretical parts of mathematics and thus do not choose to learn much theoretical mathematics. But there are special cases. PST —Preceding unsigned comment added by Point-set topologist (talk • contribs) 16:05, 11 January 2009 (UTC)

Elliptic curves aren't really relevant to what the OP is doing, and what does that last paragraph have to do with anything? --Tango (talk) 17:00, 11 January 2009 (UTC)

Have a look at his links to see the relevance of that part. PST

The steepness of the graph for large (either positive or negative) values of x is determined primarily by a. For smaller values of x, the graph will change direction a lot so it's rather more complicated. You may find it helpful to write the cubic as y=a(x-u)(x-v)(x-w), then the steepness for large values of x is, again, given by a, and u, v and w are the x-intercepts. The y-intercept would be -auvw. --Tango (talk) 17:00, 11 January 2009 (UTC)

Re-write equation as:

$y=a\left(x+{\frac {b}{3a}}\right)^{3}+\left(c-{\frac {b^{2}}{3a}}\right)\left(x+{\frac {b}{3a}}\right)+e$

where e is a function of a,b,c and d that I can't be bothered to write out. Then change co-ordinates:

$v=y-e\,;\,u=\left(x+{\frac {b}{3a}}\right)$

$v=au^{3}+\left(c-{\frac {b^{2}}{3a}}\right)u$

so we have placed the cubic's centre of symmetry at the origin. Now we can see that a determines the slope of the cubic far from its centre and $\left(c-{\frac {b^{2}}{3a}}\right)$ determines the slope at its centre, and the number of turning points. Gandalf61 (talk) 17:44, 11 January 2009 (UTC)

Sounds to me like you might be interested Bézier curve and splines in general.Dmcq (talk)